Kottwitz Map on B(G)

The Kottwitz map \(\kappa_G : B(G) \to \pi_1(G)_\Gamma\) projects \(\sigma\)-conjugacy classes to the Galois coinvariants of the algebraic fundamental group. Together with the Newton map, it gives a complete combinatorial description of \(B(G)\).

Setup

Let \(F\) be a non-archimedean local field, \(\breve F\) the completion of \(F^{\mathrm{unr}}\), \(\sigma\) the Frobenius. Let \(G\) be a connected reductive group over \(F\) and \(\pi_1(G) = X_*(T)/\mathbb{Z}\Phi^\vee\) its algebraic fundamental group, with Galois group \(\Gamma\) acting.

The Kottwitz map is a canonical homomorphism \[ \kappa_G : B(G) \;\longrightarrow\; \pi_1(G)_\Gamma, \] where \(\pi_1(G)_\Gamma = \pi_1(G) / (\sigma - 1)\pi_1(G)\) are the Frobenius coinvariants.

Construction

Torus case

For \(G = T\) a torus: \(\pi_1(T) = X_*(T)\) and the Kottwitz map is the valuation: \[ \kappa_T : B(T) \to X_*(T)_\Gamma, \qquad [t] \mapsto v(t), \] where \(v(t) \in X_*(T)\) is the cocharacter \(\chi \mapsto v(\chi(t))\). This is a bijection \(B(T) \xrightarrow{\sim} X_*(T)_\Gamma\).

General \(G\)

For a connected reductive \(G\): use the abelianisation map \(G \twoheadrightarrow G/G_{\mathrm{der}}\) (a torus) and functoriality: \[ \kappa_G : B(G) \;\to\; B(G/G_{\mathrm{der}}) \;\stackrel{\kappa_T}{=}\; X_*(G/G_{\mathrm{der}})_\Gamma \;=\; \pi_1(G)_\Gamma. \]

For non-simply-connected derived groups, the construction uses a \(z\)-extension or the Borovoi-Kottwitz crossed-module formalism.

Properties

Functoriality: For \(f : G \to G'\), the diagram \(B(G) \to B(G')\), \(\pi_1(G)_\Gamma \to \pi_1(G')_\Gamma\) commutes.
Surjectivity: \(\kappa_G\) is surjective.
Compatibility with Newton map: the image of \(\nu : B(G) \to (X_*(T)_\mathbb{Q})^\Gamma/W\) satisfies \(\bar\nu = \kappa_G \pmod{\text{linear span of coroots}}\) — i.e.\ the Newton point projects to the Kottwitz image under the natural map \((X_*(T)_\mathbb{Q})^\Gamma \to \pi_1(G)_\mathbb{Q}^\Gamma\).

Kottwitz's theorem

Theorem (Kottwitz 1997). The combined map \[ (\kappa_G, \nu_G) : B(G) \;\hookrightarrow\; \pi_1(G)_\Gamma \times (X_*(T)_\mathbb{Q})^\Gamma / W \] is injective.

The image is characterised by an "admissibility" condition: \(\nu\) must dominate \(\kappa_G\) in the dominance order, and the projection of \(\nu\) to \(\pi_1(G)_\Gamma \otimes \mathbb{Q}\) must equal \(\kappa_G\).

This is the Kottwitz classification theorem of \(B(G)\).

Examples

\(\mathrm{GL}_n\)

\(\pi_1(\mathrm{GL}_n) = \mathbb{Z}\), Galois acts trivially, so \(\pi_1(\mathrm{GL}_n)_\Gamma = \mathbb{Z}\).

\(\kappa_{\mathrm{GL}_n} : B(\mathrm{GL}_n) \to \mathbb{Z}\) sends an isocrystal of dimension \(n\) and slope sum \(s\) (i.e.\ Newton polygon ending at height \(s\)) to \(s\). Equivalently, \(\kappa(b) = v(\det b)\).

\(\mathrm{SL}_n\)

\(\pi_1(\mathrm{SL}_n) = 0\), so \(\kappa\) is trivial. \(B(\mathrm{SL}_n)\) is then classified by the Newton point alone, subject to "slope sum = 0."

\(\mathrm{PGL}_n\)

\(\pi_1(\mathrm{PGL}_n) = \mathbb{Z}/n\); \(\kappa\) detects the Brauer class of the isocrystal (modulo \(n\)).

Basic strata

The basic elements of \(B(G)\) are those with \(\kappa\) Galois- trivial (in some technical sense) — equivalently, central Newton point. They are in bijection (in a precise sense) with \(\pi_1(G)_\Gamma / (\text{torsion})\) and are the "supersingular" classes.

Role for inner forms

\(\pi_0(\kappa_G) := \ker(\kappa_G \to \pi_1(\mathrm{ad}\, G)_\Gamma)\) classifies the inner forms of \(G\) over \(F\): \[ H^1(F, G_{\mathrm{ad}}) \;\cong\; \pi_1(G_{\mathrm{ad}})_\Gamma \;\hookrightarrow\; \pi_1(G)_\Gamma. \]

So each inner form of \(G\) over \(F\) corresponds to a specific basic class in \(B(G)\) — the bridge between cohomological inner-form theory and \(\sigma\)-conjugacy.

References

Kottwitz, Isocrystals with additional structure II (Compositio 1997).
Rapoport–Richartz, On the classification and specialization of F-isocrystals with additional structure (Compositio 1996).
Görtz, Affine Deligne-Lusztig varieties and the geometry of Shimura varieties (notes, 2009).